Contraction of indices

We use contraction of indices method to manipulate Tensors. However, I cannot relate that manipulation visually. We can change covariant tensor to contravariant tensor and vice versa by contracting indices with the metic tensor. Covariant tensor represent gradient and contravariant tensor basically represent tangent vector. So, using contraction we are doing something awesome but how exactly can I can see this visually. What exactly does contraction do geometrically?

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